Find the values of 'k' for which the quadratic equation kx² - 2kx + 6 = 0 has real and equal roots. Also, find the roots.

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 28(b)

Set in the CBSE Class 10 board exam of 2024, paper 430/4/1, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerQuadratic Equations

Find the values of 'k' for which the quadratic equation $k x^{2} - 2 k x + 6 = 0$ has real and equal roots. Also, find the roots.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Real and equal roots means b² − 4ac = 0.

Hint 2

Factorise 4k² − 24k and reject the value that removes the x² term. Then substitute back and solve.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

For equal roots, $D = 0$ . Here $a = k, b = - 2 k, c = 6$ , so $D = (- 2 k)^{2} - 4 (k) (6) = 4 k^{2} - 24 k = 0$ .
1 mark
$4 k (k - 6) = 0 \Rightarrow k = 0$ or $k = 6$ . Since $k = 0$ makes the equation non-quadratic, $k = 6$ .
1 mark
With $k = 6$ : $6 x^{2} - 12 x + 6 = 0 \Rightarrow x^{2} - 2 x + 1 = 0 \Rightarrow (x - 1)^{2} = 0 \Rightarrow x = 1, 1$ .
1 mark

Answer: k = 6; roots are 1, 1

Where the marks are

Sets D = 0 to get 4k² − 24k = 0 (1 mark)
k = 6, rejecting k = 0 (1 mark)
Finds the roots x = 1, 1 (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Nature of Roots and Discriminant

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2024: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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